
Every linear array with \(N\)  elements \((N\ge 2M)\)  and \(2M\)  elements with a constant spacing \(d\)  (for example \(\lambda /2\) ) is strongly \(M\) -regular in an appropriate region \(\Omega \subset V^M\) .
This is because every \(2M\)  vectors are linearly independent if they are already independent in the \(2M\)  components of a uniform grid. For these \(2M\)  components, the matrix
\(\left(\begin{IEEEeqnarraybox}[][c]{c^{\prime }c^{\prime }c}1 &\ldots &1\\e^{-jdu_1} &\ldots &e^{-jdu_{2M}}\\e^{-j2du_1} &\ldots &e^{-j2du_{2M}}\\\vdots &\ddots &\vdots \\e^{-j(2M-1)du_1} &\ldots &e^{-j(2M-1)du_{2M}}\end{IEEEeqnarraybox}\right)\) 
must be regular. This is a van der Monde matrix of the form
\(\left(\begin{IEEEeqnarraybox}[][c]{c^{\prime }c^{\prime }c}1 &\ldots &1\\\gamma _1 &\ldots &\gamma _{2M}\\\vdots &\ddots &\vdots \\\gamma _1^{2M-1} &\ldots &\gamma _{2M}^{2M-1}\end{IEEEeqnarraybox}\right)\) 
for \(\gamma _i=e^{-jdu_i}\) , which must be regular if \(\gamma _i\ne \gamma _k\)  for all \(i\ne k\) . This means that within a period, according to (4.1-4), the vectors are linearly independent.

Every planar-element arrangement with \(N\ge (2M)^2\)  that has \(2N\times 2M\)  elements on a rectangular grid is strongly regular in a suitable region of \(\Omega \subset V^M\) .
To show this, it suffices to show the linear independence of the \(\mathbf {a}\) -vectors with respect to the grid points. Then one can write
\(\mathbf {a}(u,v)=&\left(\begin{IEEEeqnarraybox}[][c]{c}e^{-jx_1u}\left(\begin{IEEEeqnarraybox}[][c]{c}e^{-jy_1v}\\\vdots \\e^{-jy_{2M}v}\end{IEEEeqnarraybox}\right)\\\hline \\\vdots \\\hline \\e^{-jx_{2M}u}\left(\begin{IEEEeqnarraybox}[][c]{c}e^{-jy_1v}\\\vdots \\e^{-jy_{2M}v}\end{IEEEeqnarraybox}\right)\end{IEEEeqnarraybox}\right)\\=&\mathbf {a}_{(x)}(u)\otimes \mathbf {a}_{(y)}(v)\) 

if
\(\mathbf {a}_{(x)}(u)=\left(\begin{IEEEeqnarraybox}[][c]{c}e^{-jx_1u}\\\vdots \\e^{-jx_{2M}u}\end{IEEEeqnarraybox}\right)\) 
and \(\otimes \)  is the Kronecker product (tensor product) operator.
Let \(\omega _1\ldots \omega _{2M}\)  be pairwise different, which means that \(u_1,\ldots , u_r\)  are pairwise different and \( v_r\ldots v_{2M}\)  are pairwise different for \(1\le r\le 2M\) . Then, according to the first example of strongly \(M\) -regular arrays, \(\mathbf {a}_{(x)}(u_1)\ldots \mathbf {a}_{(x)}(u_r)\)  are linearly independent and \(\mathbf {a}_{(y)}(v_r),\ldots ,\mathbf {a}_{(y)}(v_{2M})\)  are linearly independent. Furthermore, the \(r\times (2M-r)\)  direction vectors \((\mathbf {a}_{(x)}(u_i)\otimes \mathbf {a}_{(y)}(v_k))\)  for \(i=1,\ldots , r\)  and \(k=r,\ldots , 2M\)  are linearly independent, because
\(\det \left(\left(\mathbf {A}_{(x)}\otimes \mathbf {A}_{(y)}\right)^*\left(\mathbf {A}_{(x)}\otimes \mathbf {A}_{(y)}\right)\right)=&\det \left(\left(\mathbf {A}^*_{(x)}\mathbf {A}_{(x)}\right)\otimes \left(\mathbf {A}^*_{(y)}\mathbf {A}_{(y)}\right)\right)\\=&\det \left(\mathbf {A}^*_{(x)}\mathbf {A}_{(x)}\right)^r\det \left(\mathbf {A}^*_{(y)}\mathbf {A}_{(y)}\right)^{2M-r}\\\ne &0\) 

as proven, for example, in [1]}, using the abbreviations
\(\mathbf {A}_{(x)}=&\left(\mathbf {a}_{(x)}(u_1)\ldots \mathbf {a}_{(x)}(u_r)\right)\\\mathbf {A}_{(y)}=&\left(\mathbf {a}_{(y)}(v_r)\ldots \mathbf {a}_{(x)}(v_{2M})\right)\) 

and thus, \(\mathbf {A}(\omega _1),\ldots ,\mathbf {A}(\omega _{2M})\)  are linearly independent.

